Math Problem Statement

Evaluate, in spherical coordinates, the triple integral off(ρ,θ,ϕ)=cosϕ, over the region0≤θ≤2π,π/3≤ϕπ/2,3≤ρ≤7integral=

Solution

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Math Problem Analysis

Mathematical Concepts

Triple Integral
Spherical Coordinates
Volume Integration

Formulas

Triple integral in spherical coordinates: ∫∫∫ f(ρ,θ,ϕ) ρ² sinϕ dρ dθ dϕ
Volume element in spherical coordinates: dV = ρ² sinϕ dρ dθ dϕ
Trigonometric identity: cosϕ sinϕ = (1/2) sin(2ϕ)

Theorems

Fundamental Theorem of Calculus
Properties of Trigonometric Functions

Suitable Grade Level

Grades 11-12 (Calculus)